If Marvel went into the magical realm eventually, and Mandarin was still around, then perhaps Iron Man could face Fin Fang Foom. Would it be that different to him luring that Leviathan in the Avengers? Of course, they probably wouldn't use Foom's name, unless Stark says it in jest when Mandarin actually says something different that sounds vaguely similar.

But since Mandarin doesn't seem to be Asian anyway, then would there really be a connection in these movies?

There can definitely be a connection between the Mandarin and Fin Fang Foom. The cosmic dragon's movie origin can diverge from that of his comics counterpart, just as the Mandarin has a different origin in the MCU. Even if they keep the detail of Fin having crash-landed in ancient China, there is no reason why he and Mandarin couldn't have encountered each other. People do travel, after all, and one would assume that alien reptiles do as well.